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Proceedings Article, Paper
@InProceedings
Beitrag in Tagungsband, Workshop

Author, Editor
Author(s):
Mütze, Torsten
Rast, Thomas
Spöhel, Reto
dblp
dblp
dblp
Not MPG Author(s):
Mütze, Torsten
Rast, Thomas
Editor(s):
Randall, Danadblp
Not MPII Editor(s):
Randall, Dana
BibTeX cite key*:
Spoehel2011
Title, Booktitle
Title*:
Coloring random graphs online without creating monochromatic subgraphs
Booktitle*:
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011)
Event, URLs
Conference URL::
http://www.siam.org/meetings/da11/
Downloading URL:
http://www.siam.org/proceedings/soda/2011/SODA11_013_muetzet.pdf
Event Address*:
San Francisco, CA
Language:
English
Event Date*
(no longer used):
Organization:
Association for Computing Machinery (ACM), Society for Industrial and Applied Mathematics (SIAM)
Event Start Date:
23 January 2011
Event End Date:
25 January 2011
Publisher
Name*:
SIAM
URL:
http://www.siam.org/
Address*:
Philadelphia, Pa.
Type:
Vol, No, Year, pp.
Series:
Volume:
Number:
Month:
January
Pages:
145-158
Year*:
2011
VG Wort Pages:
40
ISBN/ISSN:
Sequence Number:
DOI:
Note, Abstract, ©
(LaTeX) Abstract:
Consider the following generalized notion of graph coloring: a coloring of the vertices of a graph~$G$ is \emph{valid} \wrt some given graph~$F$ if there is no copy of $F$ in $G$ whose vertices all receive the same color. We study the problem of computing valid colorings of the binomial random graph~$\Gnp$ on~$n$ vertices with edge probability~$p=p(n)$ in the following online setting: the vertices of an initially hidden instance of $\Gnp$ are revealed one by one (together with all edges leading to previously revealed vertices) and have to be colored immediately and irrevocably with one of $r$ available colors.

It is known that for any fixed graph $F$ and any fixed integer $r\geq 2$ this problem has a threshold $p_0(F,r,n)$ in the following sense: For any function $p(n) = o(p_0)$ there is a strategy that \aas (asymptotically almost surely, i.e., with probability tending to 1 as $n$ tends to infinity) finds an $r$-coloring of $\Gnp$ that is valid \wrt $F$ online, and for any function~$p(n)=\omega(p_0)$ \emph{any} online strategy will \aas fail to do so.

In this work we establish a general correspondence between this probabilistic problem and a deterministic two-player game in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. This characterization allows us to compute, for any $F$ and $r$, a value $\gamma=\gamma(F,r)$ such that the threshold of the probabilistic problem is given by $p_0(F,r,n)=n^{-\gamma}$. Our approach yields polynomial-time coloring algorithms that \aas find valid colorings of $\Gnp$ online in the entire regime below the respective thresholds, i.e., for any $p(n) = o(n^{-\gamma})$.
Keywords:
random graph, graph coloring, online algorithm, average-case analysis
Download
Access Level:
Public

Correlation
MPG Unit:
Max-Planck-Institut für Informatik
MPG Subunit:
Algorithms and Complexity Group
Appearance:
MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort



BibTeX Entry:

@INPROCEEDINGS{Spoehel2011,
AUTHOR = {M{\"u}tze, Torsten and Rast, Thomas and Sp{\"o}hel, Reto},
EDITOR = {Randall, Dana},
TITLE = {Coloring random graphs online without creating monochromatic subgraphs},
BOOKTITLE = {Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011)},
PUBLISHER = {SIAM},
YEAR = {2011},
ORGANIZATION = {Association for Computing Machinery (ACM), Society for Industrial and Applied Mathematics (SIAM)},
PAGES = {145--158},
ADDRESS = {San Francisco, CA},
MONTH = {January},
}


Entry last modified by Anja Becker, 02/13/2012
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Editor(s)
[Library]
Created
01/14/2011 07:45:59 PM
Revisions
2.
1.
0.

Editor(s)
Anja Becker
Reto Spöhel
Reto Spöhel

Edit Dates
13.02.2012 14:29:31
14.01.2011 20:17:53
14.01.2011 19:46:00